this is the form given in A128729 [1]}, and \(z\)  counts only the half-length of the skew Dyck path,
and Maple's gfun translates that into a differential equation
\(31z-8-15zS-(2z-1)(44z^3+15z^2-48z+8)S^{\prime }-z(11z^2+16z-4)(2z-1)^2S{^{\prime \prime }}=0,\) 
